Wolff, Christian C.P.

English

Research Papers in Economics

F,conomics  Letters  27  (1988)  141-143 
North-Holland 
141 
AUTOREGRESSIVE  CONDITIONAL  HETEROSCEDASTICITY 
A  Comparison  of  ARCH  and  Random  Coefficient  Models 
Christian  C.P.  WOLFF 
London  Business  School,  London  NWI,  UK 
Uniuersity  of  Chicago,  Chicago,  IL  60637,  USA 
Centre  for  Economic  Policy  Research,  London  S WI  Y,  UK 
Received  16  December  1987 
Accepted  12  February  1988 
In  this  paper  it  is  shown  that  the  popular  Autoregressive  Conditional  Heteroscedasticity  (ARCH)  models  are  closely  related  to 
more  traditional  random  coefficient  models.  It  is  demonstrated  that  for  every  ARCH  model  a  simple  random  coefficient 
model  can  be  formulated  which  implies  exactly  the  same  conditional  variance  pattern  for  the  variable  of  interest. 
1.  Introduction 
In  an  influential  paper,  Engle  (1982)  presented  a  new  class  of  stochastic  processes  called 
Autoregressive  Conditional  Heteroscedasticity  (ARCH)  processes.  ARCH  processes  are  mean  zero, 
serially  uncorrelated  processes  with  non-constant  variances.  For  these  processes  the  recent  past 
carries  information  about  the  one-period  forecast  variance. 
The  general  ARCH  model  of  the  order  p  has  the  following  conditional  probability  density 
function: 
Y, I I,-,  -  N(O,h,),  (1) 
with 
tz,=(Y,+  i  a;y;2_;.  (2) 
i=l 
Here  y,  is our  variable  of  interest,  1,-i  is  the  information  set  that  is  available  at  time  t -  1  and  the 
(Y,S are  fixed  parameters.  The  expression  (2)  for  h,  captures  the  nature  of  the  conditional  hetero- 
scedasticity  that  is allowed  for  in  the  family  of ARCH  models.  In  section  2 of  this  paper,  the  random 
coefficient  model  is  introduced.  In  section  3,  it  will  be  shown  that  the  two  classes  of  models  are 
closely  related  and  that  for  every  ARCH  model  a  simple  random  coefficient  model  can  be 
formulated  which  exhibits  exactly  the  same  conditional  variance  pattern.  Section  4  contains  some 
concluding  remarks. 
0165-1765/88/$3.50  0  1988,  Elsevier  Science  Publishers  B.V.  (North-Holland) 142  C.C.P.  Wolff  / Autoregressive  conditional heteroscedasticity 
2.  Random  coefficient  models 
Hildreth  and  Houck  (1968)  considered  the  problem  of  estimating  parameters  in  random  coeffi- 
cient  models  of  the  following  general  form: 
Yt =  0,  t  (3) 
P, =  P+u,,  U, -  NID(O,Q).  (4) 
Here  x,  is  a vector  of  explanatory  variables,  &  is  a vector  of  random  coefficients  with  mean  p  and 
variance-covariance  matrix  Q. If  the first  element  of x,  is defined  to be unity,  it is unnecessary  to add 
a disturbance  term  to (3)  since  such  a term  is implicitly  contained  in (4),  attached  to the coefficient  of 
the  constant  term.  Random  coefficient  models  can  be  used  in  a  variety  of  contexts  whenever 
parameters  can  be  assumed  to  fluctuate  randomly  about  a  fixed,  but  unknown,  mean. 
3.  Comparing  ARCH  and  random  coefficient  models 
In  the ARCH  model  presented  in eqs.  (1)  and  (2)  y,  has  the following  conditional  first  and  second 
moments: 
Ebt I L,]  =O, 
var[y,(Z,_,]  =E[_v~IZ,-i]  =ao+  5  a,-)‘:,. 
i=l 
(5) 
(6) 
Now  consider  a  specialized  version  of  the  random  coefficient  model.  The  following  restrictions  are 
imposed  on  eqs.  (3)  and  (4): 
x:=  [l’Yt~l,y,-,,...,y,-,],  (7) 
p=0,  (8) 
and  Q  is a diagonal  matrix  of  order  p +  1 with  the  following  diagonal  elements: 
Qrr=~L  i=2,3 
7...>  p+l.  (10) 
In  this  specialized  version  of  the  random  coefficient  model  y,  has  the  following  conditional  first  and 
second  moments: 
E[.~,l1,-,1  =O,  (11) 
var[y,(Z,_,]  =E[y?lZ,_,]  =a*+  i  u,~Y:,. 
i=l 
(12) 
Note  that  the  conditional  means  in  (5)  and  (11)  are  both  zero  and  that  the  conditional  variances  in C. C. P.  Worff  /  Autoregressive  conditional  heteroscedasticity  143 
(6)  and  (12)  have  the  same  linear  structure.  If  we  set  (Ye  =  0’  and  LY,  =  a:  for  i =  1, 2,.  . _, p,  the 
ARCH  model  has  exactly  the  same  conditional  variance  as  the  random  coefficient  model. 
4.  Concluding  remarks 
In  this  paper  we  have  shown  that  the  popular  ARCH  models  are  closely  related  to  traditional 
random  coefficient  models.  For  every  ARCH  model  a random  coefficient  model  can  be  found  which 
implies  exactly  the  same  pattern  for  the  conditional  variance  of  the  variable  of  interest. 
Engle  (1982)  shows  that  the  pth  order  ARCH  model  satisfies  certain  regularity  conditions  if 
I.Q > 0  and  OL,,  OLD,.  . . , (Ye  2  0.  These  inequalities  are  automatically  satisfied  in  the  context  of  the 
random  coefficient  model. 
References 
Engle,  Robert  F.,  1982,  Autoregressive  conditional  heteroscedasticity  with  estimates  of  the  variance  of  United  Kingdom 
inflation,  Econometrica  50,  987-1007. 
Hildreth,  Clifford  and  James  P.  Houck,  1968,  Some  estimators  for  a  linear  model  with  random  coefficients,  Journal  of  the 
American  Statistical  Association  63,  584-595. 

Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation,

Some estimators for a linear model with random coefficients,

Autoregressive conditional heteroscedasticity. A Comparison of ARCH and Random Coefficient Models.

http://arno.unimaas.nl/show.cgi?fid=10624

Autoregressive conditional heteroscedasticity. A Comparison of ARCH and Random Coefficient Models.

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